The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts​ (a) through​ (c).a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

Question
Asked Sep 23, 2019
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The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts​ (a) through​ (c).

a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

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Expert Answer

Step 1

A normal distribution can be visualized by looking at the curve below. 

X-axis shows Z score which is the distance from mean in terms of standard deviation. A z score of 2 would mean that the distance of the data point from the mean is equal to 2 standard deviation.

Commute time within 2 standard deviations of the mean would mean all communte times between Z score of -2 and +2 as shown by the shaded portion under the curve.

+2
+1
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+2 +1

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Step 2

The area under the curve represents probability or percentage or proportion. Since the value of probability ranges from 0 to1, the total area under the curve is always equal to 1 or 100%. 

To calculate area percentage or probability corresponding to any Z score , Z tables can be used which yields percentage of area to the left of a score

We are interested in P(-2<Z<2) which will be equal to P(Z<2) - P(Z<-2) 

This can be represented graphically as shown below.

 

+2
+1
P(Z<2)
P(Z<-2)
P(Z<2)-P(Z<-2)
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+2 +1 P(Z<2) P(Z<-2) P(Z<2)-P(Z<-2)

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Step 3

Let's see how Z table can be read for a Z score of -2 and +2

To find P(Z<=-2) , we have to find the value that lies at the intersection of row -2 and column .00

(NOTE: If this was P(Z<=-1.57), then we would look for value at the intersec...

00
.01
.02
.03
.04
.05
.06
.07
.08
.09
Z.
0003
.0003
0002
-3.4
0003
0003
.0003
0003
0003
0003
0003
0005
.0004
0003
3.3
0005
0005
0004
.0004
0004
0004
.0004
0005
-3.2
.0007
0006
.0006
.0006
0006
.0005
0005
0007
.0006
0007
0009
0008
0008
.0008
.0007
3.1
0010
0009
0013
0009
0008
0013
0012
0010
.0013
0012 .0011
0011
.0011
.0010
-3.0
0015
.0015
0014
0014
2.9
0019
0018
0018
0017
.0016
0016
0021
.0029
-2.8
0025
0024
0023
.0021
0026
0023
0022
0020
0019
0030
-2.7
.0028
0027
.0026
0035
0034
0033
0032
0031
0041
-2.6
0044
0039
.0047
0045
0043
0040
.0038
0037
0036
.0049
0057
.0055
0054
0052
.0051
0048
-2.5
0062
0060
0059
-2.4
0071
0082
.0080
0078
0075
0073
0069
.0068
.0066
.0064
0091
0099
.0096
0094
.0089
.0087
0084
2.3
0107
.0104
0102
-2.2
0132
.0113
0110
.0139
.0136
0129
.0125
.0122
0119
0116
0179
-2.0 .0228
0174
.0166
.0162
.0158
0154
.0150
0146
0143
2.1
0170
.0192
0188
0183
0222
0217
.0212
0207
0202
0197
0274
0268
.0262
0256
0250
0239
0233
0287
0281
.0244
-1.9
0344
.0314
0307
0301
0294
-1.8
0359
0351
0336
0329
0322
0367
-1.7
0446
0436
0427
0418
0409
0401
0392
0384
0375
0495
0485
0475
0465
0455
-1.6
0548
0537
0526
0516
0505
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00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Z. 0003 .0003 0002 -3.4 0003 0003 .0003 0003 0003 0003 0003 0005 .0004 0003 3.3 0005 0005 0004 .0004 0004 0004 .0004 0005 -3.2 .0007 0006 .0006 .0006 0006 .0005 0005 0007 .0006 0007 0009 0008 0008 .0008 .0007 3.1 0010 0009 0013 0009 0008 0013 0012 0010 .0013 0012 .0011 0011 .0011 .0010 -3.0 0015 .0015 0014 0014 2.9 0019 0018 0018 0017 .0016 0016 0021 .0029 -2.8 0025 0024 0023 .0021 0026 0023 0022 0020 0019 0030 -2.7 .0028 0027 .0026 0035 0034 0033 0032 0031 0041 -2.6 0044 0039 .0047 0045 0043 0040 .0038 0037 0036 .0049 0057 .0055 0054 0052 .0051 0048 -2.5 0062 0060 0059 -2.4 0071 0082 .0080 0078 0075 0073 0069 .0068 .0066 .0064 0091 0099 .0096 0094 .0089 .0087 0084 2.3 0107 .0104 0102 -2.2 0132 .0113 0110 .0139 .0136 0129 .0125 .0122 0119 0116 0179 -2.0 .0228 0174 .0166 .0162 .0158 0154 .0150 0146 0143 2.1 0170 .0192 0188 0183 0222 0217 .0212 0207 0202 0197 0274 0268 .0262 0256 0250 0239 0233 0287 0281 .0244 -1.9 0344 .0314 0307 0301 0294 -1.8 0359 0351 0336 0329 0322 0367 -1.7 0446 0436 0427 0418 0409 0401 0392 0384 0375 0495 0485 0475 0465 0455 -1.6 0548 0537 0526 0516 0505

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